# Romberg array Calculator

## Calculates the Romberg array produced by applying Richardson’s extrapolation procedure, which forms the basis of the Romberg Method.

 The final diagonal element so produced is the best estimate of the integral.$I={\large\int_a^{\hspace{25}b}}f(x)dx=R_{\small n}^{\small\ k}+O\((\frac{b-a}{2^n})^{\small 2k+2}\)\\[10]\hspace{10} R_{\small 0}^{\small\ 0}\\\hspace{10} R_{\small 1}^{\small\ 0}\hspace{30} R_{\small 1}^{\small\ 1}\\[10]\hspace{10} \cdots\hspace{40} \cdots\hspace{30} \ddots \\[10]\hspace{10} R_{\small n-1}^{\small\ 0}\hspace{15} R_{\small n-1}^{\small\ 1}\hspace{15} \cdots \hspace{10} R_{\small n-1}^{\small\ n-1}\\\hspace{10} R_{\small n}^{\small\ 0}\hspace{30} R_{\small n}^{\small\ 1}\hspace{30} \cdots \hspace{10} R_{\small n}^{\small\ n-1}\hspace{20} R_{\small n}^{\small\ n}\\$
f(x)
a
 , b
 maximum step n 2 3 4 5 6 7 8 9 10 partitions N=2n
rule
 Trapezoidal Midpoint Non-linear substitution (Midpoint)
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt

 The Romberg integration Rnn can be rapidly converged with the increase in n.It is calculated by increasing the number of partitions to double from 2 to N=2n.In the case of non-analytic at endpoints of f(x), you can calculate by Midpoint rule.In the case of a periodic function, you can calculate by Non-linear substitution in x. $\normal Romberg\ integration\\[10](1)\ Trapezoidal\ rule\\\hspace{10} R_{\small 0}^{\small\ 0}= {\large\frac{h_0}{2}}\{f(a)+f(b)\}, \hspace{20}h_0=b-a\\\hspace{8} R_{\small n}^{\small\ 0}= {\large\frac{R_{\small n-1}^{\small\ 0}}{2}}+h_n{\large \sum_{\tiny j=1}^{\small 2^{n-1}}}f(a+(2j-1)h_n)\\\hspace{190}h_n={\large\frac{b-a}{2^n}}\\[10](2)\ Midpoint\ rule\\\hspace{10} R_{\small 0}^{\small\ 0}= h_0f(a+{\large\frac{h_0}{2}}),\hspace{20}h_0=b-a\\\hspace{10} R_{\small n}^{\small\ 0}= h_n{\large \sum_{\tiny j=1}^{\small 2^n}}f(a+(j-\frac{1}{2})h_n), \hspace{15}h_n={\large\frac{b-a}{2^n}}\\(3)\ R_{\small n}^{\small\ k}= {\large\frac{4^{\small k} R_{\small n}^{\small\ k-1}-R_{\small n-1}^{\small\ k-1}}{4^{\small k}-1}}\\[10](4)\ Relative\ Error\hspace{30} \epsilon_{\small n}=\left|\frac{R_{\small n}^{\small\ n}-R_{\small n-1}^{\small\ n-1}}{R_{\small n}^{\small\ n}}\right|\\[30]Non-linear\ substitution\ in\ x\\\hspace{25} I={\large\int_a^{\hspace{25}b}}f(x)dx\ ={\large\int_{\small-1}^{\hspace{25}\small1}}f(x)\ \frac{b-a}{2}\ \frac{3(1-u^2)}{2}du\\\hspace{100}x=\frac{b-a}{2}t+\frac{b+a}{2},\hspace{20}t=\frac{u}{2}(3-u^2)$

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